Journal of Composite Materials - Semantic Scholar

Journal of Composite Materials http://jcm.sagepub.com

Finite Element Based Energy Dissipation Studies of Al-SiC Composites Narasimalu Srikanth, Khine Khine TUN and Manoj Gupta Journal of Composite Materials 2003; 37; 1385 DOI: 10.1177/0021998303034362 The online version of this article can be found at: http://jcm.sagepub.com/cgi/content/abstract/37/15/1385

Published by: http://www.sagepublications.com

On behalf of: American Society for Composites

Additional services and information for Journal of Composite Materials can be found at: Email Alerts: http://jcm.sagepub.com/cgi/alerts Subscriptions: http://jcm.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.com/journalsPermissions.nav Citations (this article cites 24 articles hosted on the SAGE Journals Online and HighWire Press platforms): http://jcm.sagepub.com/cgi/content/refs/37/15/1385

Downloaded from http://jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 10, 2008 © 2003 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

Finite Element Based Energy Dissipation Studies of Al–SiC Composites NARASIMALU SRIKANTH,* KHINE KHINE TUN AND MANOJ GUPTA Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore – 117576 (Received July 11, 2002) (Revised February 6, 2003)

ABSTRACT: In the present study, the damping capacity of Al–SiC composite formulations is predicted using a visco-plasticity based micro-mechanical modelling approach. The model is based on finite element analysis of an axisymmetric unit cell, which mimics a pure Al cylinder with a spherical SiC reinforcing particulate placed at the centre. The energy dissipated by the composite is numerically predicted using the unit cell by applying a harmonic load, taking into account the viscous behavior of the processing induced residual plastic strain at the matrix–reinforcement interface of the composite. The model shows that the plastic zone size increases with volume fraction of SiC added resulting in a proportional increase in damping capacity of the composite. The model was validated by comparing the numerical results against an impact based suspended beam experiment conducted at low strain amplitude on Al–SiC samples with different volume fraction of SiC particulates. In addition, an attempt has been made to study the effect of particulates’ stress concentration features as well as process-induced defects on the overall damping capacity of the composite. KEY WORDS: plastic zone, damping coefficient, loss factor, anelasticity, composite.

INTRODUCTION ETAL MATRIX COMPOSITES (MMCs) forms a new breed of structural materials in the engineering community. Particulate type reinforcement provides isotropic material behavior in a global perspective by combining the metallic properties of the matrices with ceramic properties of the reinforcements leading to greater stiffness, strength, hardness and thermal stability. Studies of various particulate reinforced material systems also revealed that the particulate reinforcement enhances the damping capacity of the metallic

M

*Author to whom correspondence should be addressed. E-mail: [email protected]

Journal of COMPOSITE MATERIALS, Vol. 35, No. 15/2003 0021-9983/03/15 1385–26 $10.00/0 DOI: 10.1177/002199803034362 ß 2003 Sage Publications Downloaded from http://jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 10, 2008 © 2003 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

1385

1386

N. SRIKANTH ET AL.

matrix [1–3] and as a result they are becoming one of the most potential and promising candidates in automobile, aerospace, electronics and sports sectors. The present study aims to rationalize the improvement in damping capacity in terms of the residual strain at the particulate–matrix interface, in addition to the other prevailing damping mechanisms in the metallic matrix, such as point defects relaxation, dislocation motion, grain boundary sliding, elastothermodynamic and magnetoelastic effects [4–6]. Literature review shows finite element method is a useful numerical tool capable of predicting the thermophysical properties of MMCs [7–10]. Research findings of Xu and Schmauder [11] and Wang et al. [12] have been successful in studying the plastic energy dissipation in MMCs at high strain amplitudes. Their model was based on an elasticplastic model and did not take in to account the residual stresses in the composite. Hence their prediction at low strain amplitude showed insignificant damping capacity for the MMC which is contrary to experiment results. At high strain amplitude due to constrained behavior at the metal–particulate interface the metallic matrix yields and hence their model could predict significant damping. Hence in the present study, presence of residual stresses in the MMCs, when cooling down from the extrusion temperature to room temperature, which is well established by experiments [13], have been accounted in the energy dissipation studies of the composite. Considering the significant potential of aluminium due to its higher specific stiffness along with excellent castability and machinability, the present study focuses on the enhancement of the damping capacity due to the presence of SiC particulates. The theoretical predictions are made using a finite element method (FEM) based unit cell model (UCM) that uses a rate-dependent based plasticity model. The numerical predictions were further validated using experimental results obtained from suspended beam technique coupled with circle-fit approach [2]. In addition, the present study also addresses the applicability of the UCM to study the effect of process-induced defects such as voids, particulate breakage and metal–ceramic interface de-bondage on the overall damping capacity of the composite.

THEORETICAL BACKGROUND Damping Capacity of Material In any vibrating material, some energy loss is inherent which is generally referred to as passive damping. If the energy loss is due to inherent nature of the material it is measured in terms of loss factor , which is proportional to the ratio of the energy lost per cycle W to the maximum potential energy stored in the system ‘U’ during that cycle and is defined as follows [4]: ¼

W 2U

ð1Þ

Metallic materials when subjected to cyclic deformation are known to undergo energy dissipation forming a hysteresis loop, which is a useful concept since the energy lost per cycle is related to the area enclosed by the hysteresis loop (see Figure 1(f )). This concept is quite useful when non-linear damping mechanisms exist in the material, which arises from several mechanisms as reported by Lazan [4]. Among the important non-linear mechanisms


Finite Element Based Energy Dissipation Studies of Al–SiC Composites

1387

Figure 1. Methodology for deducing a simplified FEM model for the PRMMC: (a) hexagonal representation of a particulate reinforced MMC; (b) unit hexagonal cell; (c) approximated cylindrical unit cell; (d) axisymmetric FEM model of the cylindrical unit cell; (e) equivalent stress contour plot and (f) hysterisis curve.

is ‘‘plastic strain’’ damping. This mechanism does not require plastic strain throughout the specimen, but it is assumed that the plastic strain occurs on a microscopic scale due to inhomogenities within the matrix material. Even for a monolithic material, on the microscopic scale, the inhomogeneity of stress distribution within crystals and the stress concentration at crystal boundary intersections produce local stress high enough to cause local plastic strain, even though the average (macroscopic) stress may be very low [5]. Consider the typical stress–strain hysteresis loop for a non-linear material shown in Figure 1(f ). For plastic strain damping, the energy lost per cycle, W, is the energy absorbed by the material which is proportional to the area enclosed by the hysteresis loop and is given by [5]: W ¼ Jan

ð2Þ

where J is the damping energy at stress amplitude of unity, a is the stress amplitude and n is the damping exponent. Based on elementary strain energy concepts, the damping loss factor can be expressed as follows [5]: ¼

EJam

ð3Þ

where E is the elastic modulus and m is equal to (n 2) and is a material property; however, this property is dependent on the stress amplitude and the material’s stress


1388

N. SRIKANTH ET AL.

history. For metallic materials, m is small between zero and one, for stress amplitudes less than the fatigue strength but may be twenty or greater for stresses above the fatigue strength, depending on the stress history [5]. Based on works of Goodman [5], below a peak stress, L, known as the ‘‘cyclic stress sensitivity limit’’, the curve of damping versus stress amplitude is a straight line on a log–log plot and displays no stress-history effect. The limit L usually falls somewhat below the fatigue strength of the material. Above L, stress-history effects appear. It may be noted that under low damping conditions such as in composite specimens under low strain amplitude, Equation (4) can be used to relate the loss factor to the other damping measures, where, is the ratio between the energy dissipated during each loading cycle, W, and the maximum energy stored during the cycle, U, is the damping ratio, is the logarithmic decrement, is the phase angle between the applied stress and the resultant strain, Q1 is the inverse quality factor, A(t1) and A(t2) are the displacements at times t1 and t2, respectively, and fr is the resonant frequency [6]. ¼ W=U ¼ 4 ¼ 2 ¼ 2 tan ¼ 2 ¼ 2Q1 ¼

2 lnðAðt1 Þ=Aðt2 ÞÞ fr ½t2 t1

ð4Þ

FINITE ELEMENT METHOD Finite Element Method is an efficient tool to analyse particulate dispersed MMCs due to its modelling capability of the particulate. FEM is one of the tools, which has the distinct advantage to model more closely to the exact geometry. Secondly, it has the advantage to model the various material properties in the geometry as well as the anisotropy of the individual phases. Thirdly, it can model the exact fixity and the load conditions on the specimen [14]. Mostly FEM has been successfully used in analysing continuous and discontinuous reinforced composites. In the present work, a cell model approach is adopted to study the energy dissipation capability of the particulate reinforced composites. The composite material was idealised in terms of periodic array of identical hexagonal cylindrical cells, shown in Figure 1(a), with the SiC reinforcement located in the centre of each cell. Axisymmetric cylindrical cells, which can be regarded as approximations to the threedimensional array of the hexagonal cylindrical cells [7–11], were used in the calculation (see Figure 1(c)).

Unit Cell Model Exact micro-mechanical modelling of a two or more phase composite material is very complicated due to the geometric size and shape of the individual phases. Hence UCMs have been designed and successfully used by a number of researchers, including Llocra [7], Bao [8], and Shen et al. [9] to calculate the thermo-mechanical properties of composites. These models were successfully used to predict results similar to more rigorous and computationally more intense three-dimensional models. In the present study, an axisymmetric cell model similar to Bao [8] is chosen to model a particulate reinforced



1389

MMC. This limits the particulate shape to axially symmetric in nature such as a sphere, cylinder, or an ellipsoid and provides the convenience to analyse the 3D model as a 2D axi-symmetric model, which results in high accuracy with least computer resource. The UCM for a spherical particulate is shown in Figure 1(d). Cylindrical coordinates are defined so that r represents the radial direction and z represents the axial direction of the cylinder. In the present study, the particulate is assumed to be a spheroid, which occupies the region r2 þ z2 R2, where R is the particulate radius. The particulate is placed at the exact centre of the metal matrix cylinder with a radius L and length 2L. The dimensions of the cylinder are so chosen so as to satisfy the required volume fraction of the reinforcement in the composite under study. The reinforcement volume fraction was taken as the ratio of the reinforcement volume to the cell volume. The aspect ratio of the cell, the ratio of the height of the cylinder to its diameter, was taken to be the same as that of the reinforcement. The cylindrical surface is constrained to remain cylindrical but can move in or out with zero average normal traction. The faces perpendicular to the direction of stressing also remain planar with zero shear traction and a sinusoidally varying normal stress. Before applying such a cyclic load, the UCM was subjected to thermal cooling from extrusion temperature to room temperature to account for the residual stress in the subsequent calculations. The ANSYSTM code was employed using 8-noded viscoplastic (Visco108) elements to solve the boundary value problem formulated above [15]. Figure 1(a)–(f) shows the overall modelling methodology of particulate reinforced MMCs. Constitutive Behavior The maximum temperature experienced by the SiC particulate, during extrusion process, is significantly below the melting temperature, hence we assume that the particulate obeys a thermoelastic material behavior. However, for the aluminium matrix, the extrusion temperature corresponds to a homologous temperature equal to 0.67 during extrusion process. In the previous study of Xu and Schmauder [11] and Wang et al. [12], the matrix was assumed as elasto-plastic and without any process induced residual plastic strain. While in the present study, the metallic matrix is assumed to be a thermo-elastoviscoplastic material behavior, whose state is described by a structural parameter s, in addition to the stress and temperature T, to account for the isotropic work-hardening and recovery [16]. To specify the constitutive equation of the metallic matrix, the rate-dependent plasticity (or visco-plasticity) is characterized by the irreversible straining that occurs in the material over time. The plastic strains are assumed to develop as a function of the strain rate. Here we present the rate-dependent model developed by Anand [16]. This rate-dependent model differs from the rate-independent model in that there is no explicit yield condition, and no loading/unloading criterion is used. Instead, plastic flow is assumed to take place at all non-zero stress values, although at low stresses the rate of plastic flow may be immeasurably small. The total strain rate tensor D is the sum of the elastic strain rate tensor De, thermal strain rate tensor Dth (which is equal to T m ðuÞ, where (u) is the second-order unit tensor and T is the temperature and the dot in the superscript refer to the rate of change with time), viscoplastic strain rate tensor Dp, as follows: D ¼ De þ Dth þ Dp


ð5Þ

1390

N. SRIKANTH ET AL.

The viscoplastic deformation is assumed to be isochoric process (i.e., volume is constant). Based on small strain theory [16], the evolution equation for the cauchy stress is given as follows: ¼ 2 M ðD Dp Þ þ ðKM 2 M =3ÞðtrDÞðuÞ 3KM M T ðuÞ

ð6Þ

where M and KM are the elastic shear and bulk moduli of the metallic matrix. The 0 viscoplastic strain rate is related to the stress deviator as follows: Dp ¼ 3 0 d p =2

ð7Þ

where, d p is the equivalent plastic strain rate and is the von-mises equivalent tensile stress in terms of deviatoric stress 0 as follows: ¼ ð3=2 0 : 0 Þ0:5

ð8Þ

Further, the equivalent plastic strain rate, which is determined by the consistency condition in the rate-independent model, needs to be prescribed by an appropriate constitutive function. The specifics of this constitutive equation are the flow equation: d p ¼ AeQ=R ½sinhð =sÞ1=m

ð9Þ

where A is the pre-exponential factor, is the absolute temperature, R is the universal gas constant, Q is the activation energy, m is the strain rate sensitivity index, is multiplier of stress, and s is the structural parameter which corresponds to the evolution of the deformation resistance and is given in the rate form ðs Þ as follows: s ¼ ½h0 ðjBjÞa sgnðBÞd p

ð10Þ

where h0 is hardening constant, a is the strain-rate sensitivity index for both hardening and softening and sgn (B) refers to the sign of the parameter B which is a function of s as follows: B ¼ 1 s=s

ð11Þ

where s refers to the saturation value of deformation resistance which is expressed as follows: p n d expðQ=R Þ s ¼ s^ A

ð12Þ

where s^ is the coefficient for deformation resistance saturation value, n is the strain rate sensitivity parameter of saturation (deformation resistance) value. Equation (10) allows to model for both strain hardening and strain softening material behaviors. Based on the works of Brown et al. [17], for most of the common deformation the static recovery phenomena is negligible and hence neglected in the above formulation.



1391

Numerical Implementation Based on the above model along with a consistent stress update procedure, which is equivalent to the Euler backward scheme to enforce the consistency condition, analysis was conducted. The evolution equation (10) at the end of the time step and the ratedependent and rate-independent plasticity are computed accordingly [15]. Further, the plastic work done EePl in each element is computed for each time step as follows: EePl ¼

NINT X NCS X i¼1

fgT f"pl gvoli

ð13Þ

j¼1

where NINT and NCS refers to the number of integration points within the element and number of iterations, respectively, parameter voli referto the volume of ith element, f g is the stress vector, "el is the elastic strain and "pl is the plastic strain increment. A programme was written using Ansys Parametric Design Language (APDL) [15] to calculate the accumulated plastic energy (W) in one stress cycle. The maximum strain energy was computed using the predicted values of the elastic modulus of the MMC, Emmc, and the maximum stress amplitude, max , and the volume of the unit cell, v, as follows: U¼

2 max v Emmc

ð14Þ

The accumulated plastic energy and the maximum strain energy were further substituted in Equation (1) to deduce the damping loss factor.

EXPERIMENTAL PROCEDURES Materials and Processes In this study, aluminium based composites containing 5.2, 13.9 and 17.7 weight percentages of SiC particulates were synthesized using mechanical disintegration and deposition (MDD) method and a pure Al sample was synthesized using conventional casting method (CCM) [18]. They were subsequently hot extruded at 350 C. Scanning electron microscopy (SEM) was used to study the distribution and interface quality of SiC in the matrix of the extruded samples. Optical microscopy was used to study the grain size and morphology of the as-processed samples.

Density and Porosity Measurement The densities of the extruded composite samples were measured by Archimedes’ principle [19]. The specimens were weighed in air and in distilled water using an A & D ER-182A electronic balance to an accuracy of 0.0001 g. The porosity was then calculated from the measured composite density, theoretically computed rule of mixture density and the reinforcement weight fractions [19].


1392

N. SRIKANTH ET AL.

Thermo-mechanical Testing The thermo-mechanical study was intended to measure the coefficient of thermal expansion (CTE). The test was carried out using the Setaram 92 TMA 16-18 machine that uses a spherical ended, 5 mm-diameter alumina probe. The top and bottom surfaces of test specimen were ground using a 300 grit SiC abrasive paper to ensure that the specimens’surfaces were flat. A test sequence of temperature range from 20 to 400 C was used for each of the extruded specimen. The temperature was measured by a K-type, coaxial thermocouple and the test was carried out in an inert environment of argon gas. The data were obtained in the form of curves of dimension changes versus temperature and time. The Setaram software was used to calculate the average coefficient of thermal expansion.

Hardness Measurement The micro-hardness measurements were conducted on a digital micro-hardness tester (Matsuzawa model MXT50). The tester makes use of a pyramid diamond indentor with a facing angle of 136 . Micro-hardness measurement was carried out using a test load of 25 gf and a dwell time of 15 s. Vickers hardness (HV) was obtained from the test load applied to form an indentation on the test sample. The test samples were polished to 5

with alumina prior to measurement. The macro-hardness of the samples was measured using the Rockwell Tester (Future– Tech model FR-3). The measurement was made using the Rockwell 15T superficial scale, which uses a 1.58 mm diameter steel ball diameter with a 15 kgf test load. The test samples were polished with 5 alumina suspension prior to measurement to provide a smooth surface free from oxide scale and foreign matters. The test surface and the bearing surface of the test pieces were maintained flat and perpendicular to the axis of the indentor to ensure accurate results. In addition, the cleanliness and smoothness of the seating and supporting surfaces of the anvil were also ensured prior to testing.

Tensile Testing and Fractography The smooth bar tensile properties of the extruded samples were determined in accordance with the ASTM E8M-96 standard [20]. The tensile tests were conducted on round tension test specimens of diameter 5 mm and gauge length 25 mm using an automated servo-hydraulic testing machine (Instron 8501), with cross head speed set at 0.254 mm min1. An initial strain rate of 1.69 104 s1 was used. Fractographic studies using SEM were also conducted on the tensile fractured surfaces of the extruded composite samples in order to provide insight into their failure mechanisms.

Damping Measurement – Suspended Beam Method The impact-based ‘‘Free–Free’’ or ‘‘Suspended’’ beam method was performed based on the ASTM C1259-98 standard [21]. Description of the experimental set-up is described in [2]. The receptance frequency response function (FRF) corresponding to the resonance



1393

Figure 2. (a) Circle-fit plot of FRF data for the pure aluminium sample and (b) Use of the natural frequency and two data points to derive the damping factor.

condition was plotted as a nyquist plot [22]. Using least square technique a circle was fit and based on hysterically damped vibrating system the circle diameter can be shown to be inversely dependent on the damping coefficient. Figure 2(a) shows a typical circle fit plot of the pure aluminium sample. The exact location and determination of the natural frequency and the corresponding damping factor ‘’ was calculated using frequency spacing technique [2]. Figure 2(b) shows the typical circle plot from which two points are selected for damping loss factor calculation, denoted as point a and b corresponding to frequencies !a and !b, respectively, which are lesser and greater than the natural frequency !n, respectively. Thus, the damping factor r can be expressed using the angles shown in Figure 2(b), as follows [2]: free ¼

!2a !2b 1 !2r tanð a Þ þ tanð b Þ

ð15Þ

RESULTS Microstructure Characterization The results of the microstructural studies conducted on the monolithic and MMC specimens are listed in Table 1. Figure 3(a) is a typical SEM micrograph showing the distribution of the SiC particulates in the aluminium matrix containing 5.2 wt.% of SiC reinforcement. Uniform distribution of SiC was found with some presence of clustering and porosity in all the three different composite samples. Close inspection of the SiC particulate at high magnification illustrates that the interfacial bonding (assessed in terms of voids and debonded regions) in the three categories of samples is good. Figure 3(b), for example, illustrates the SiC–Al interfacial integrity in the case of Al–5.2 wt.% SiC sample. The grain morphology of the extruded specimens was observed using an optical microscope and representative micrographs are shown in Figure 3(c) and (d). The grain morphology of the matrix in both unreinforced and reinforced samples was found to be equiaxed. Table 1 shows a decrease in average grain size with an increase in weight percentage of SiC.


1394

N. SRIKANTH ET AL.

Table 1. Results of microstructural studies conducted on monolithic and Al–SiC samples. SiC Weight% 0.0 5.2 13.9 17.7

Density (g/cm3)

Porosity (%)

Metal Matrix Grain Size (mm)

Grain Aspect Ratio

SiC Size (mm)

SiC Aspect Ratio

2.697 2.713 2.732 2.741

0.12 0.26 0.77 0.95

55 14 47 13 42 9 36 9

1.53 1.54 1.54 1.48

– 15 5 15 4 12 2

– 1.65 1.61 1.52

Figure 3. (a) SEM micrograph showing the distribution of SiC particulate in Al–5.2 wt .% SiC sample; (b) SEM micrograph showing the particulate–matrix interface in Al–5.2 wt. % SiC sample; (c) Optical micrograph showing equiaxed grain morphology of pure aluminium sample; (d) Optical micrograph showing equiaxed grain morphology Al–17.7 wt. % SiC sample.

Density and Porosity Measurement The results of the experimental density and porosity values are listed in Table 1. The results indicate the potential of MDD technique to synthesize near-dense materials.

Thermo-mechanical Testing The average CTE of the three composite samples are listed in Table 2. Results show that the CTE of the aluminium matrix decreases with an increase in SiC weight percentage.


1395

Finite Element Based Energy Dissipation Studies of Al–SiC Composites Table 2. Results of hardness and CTE studies of Al–SiC composite samples. Weight % of Reinforcement 0.0 5.2 13.9 17.7

Volume % of Reinforcement

Matrix Hardness (HV)

Interface Hardness (HV)

Macro Hardness (HR15T)

Coef. of Thermal Expansion exp (106/ C)

0.0 4.4 12.0 15.4

39 1.0 46 0.4 60 0.6 73 0.8

– 265 5.8 309 6.7 338 7.0

39 2 43.4 0.8 64.7 2.6 68.4 1.7

25.890 24.566 23.213 22.555

Table 3. Results of tensile testing obtained from Al and Al–SiC composite samples. Weight %

Tensile Modulus* (GPa)

Yield Strength* (MPa)

Ultimate Strength* (MPa)

Ductility* (%)

69.4 3 70.8 1 71.1 2 72.6 2

93 8 96 11 99 7 105 2

126 7 126 4 131 12 141 1

25.3 4 24.8 2 17.4 5 14.4 4

0.0 5.2 13.9 17.7

*Average values of atleast three successfully tested samples in accordance with ASTM E8M-96 standard.

Hardness Testing The results of hardness measurements carried out on MMC samples are summarized in Table 2. It may be noted that the average values of both microhardness and macrohardness of the samples increased with an increase in the weight percentage of reinforcement. The results also revealed that the interface hardness in all the three composite samples was significantly higher when compared to the matrix hardness and its average value also increased with an increase in the weight percentage of the reinforcement. Tensile Testing and Fractography The results of tensile testing (static elastic modulus, 0.2% yield strength, ultimate tensile strength and ductility) are listed in Table 3. The results revealed that an increase in the weight percentage of SiC particulates in the aluminium matrix leads to: (a) an increase in elastic modulus, (b) minimal upward change in 0.2% yield strength (YS), (c) increase in ultimate tensile strength (UTS) and (d) decrease in failure strain. Figure 4 shows typical interfacial debonding and particulate breakage observed in the fractured surface of the composite sample containing 17.7-wt.% SiC. The results of the fracture surface analysis conducted on the tensile tested specimens revealed ductile fracture in the case of pure aluminium and particulate fracture particulate–matrix interfacial debonding as the reinforcement associated fracture mechanisms in the case of composite specimens. Damping Measurement Suspended beam vibration testing was used in the present study to compute the damping loss factor. The results were benchmarked against a pure aluminium sample.


1396

N. SRIKANTH ET AL.

Figure 4. SEM micrographs showing: (a) Particulate breakage and (b) Interfacial debonding of SiC particulate in the case of Al–17.7 wt.% SiC sample.

Table 4. Experimental and numerically predicted microstructural and mechanical characteristics of Al and Al–SiC samples.

SiC wt.(%) 0.0 5.2 13.9 17.7

Modulus Prediction Eeshelby (GPa)

Volume Percentage of Plastic Zone (%)

Dislocation Density (m2)

Interparticulate Spacing (mm)

Exp. Loss Factor free

FEM Pred. Loss Factor Pred

69.4 71.3 74.8 76.5

– 29.33 76.90 98.40

– 9.87E þ 11 3.03E þ 12 4.70E þ 12

– 90.90 51.75 38.10

0.00293 0.00335 0.00453 0.00578

0.00293 0.00325 0.00449 0.00513

Table 4 lists the damping loss factor free calculated using Equation (15), for the monolithic and MMC samples.

Numerical Predictions The UCM proved to be a proper idealization of the particulate reinforced MMC. The model assumed Al as the metallic matrix and SiC as the reinforcing particulates. The elastic modulus and thermal expansion coefficient of aluminium was taken as 69.4 GPa and 25.89 106 / C, respectively, based on the present study. The elastic modulus and thermal expansion coefficient of SiC particulates were assumed to be 410 GPa and 4.3 106 / K, respectively [11]. The Poisson’s ratio of aluminium and SiC was taken to be 0.3 and 0.25, respectively [23]. The metallic matrix (Al-1100-O) was assumed to obey Anand viscoplastic material model for which the material constants were assumed as A ¼ 1.91 107 s1, ¼ 7.0, m ¼ 0.23348, s0 ¼ 18 MPa, h0 ¼ 1115.6 MPa, Q/Rg ¼ 21090 K, ¼ 1.3, n ¼ 0.07049 and s^ ¼ 18.9 MPa [15]. The cooling rate adopted to predict the thermally induced residual stress was 100 C/s [23]. The unit cell was made corresponding to a spherical particulate of size 15 mm inside a cylindrical metallic matrix with a length to diameter ratio of 1 and a volume modelled such



1397

that the volume fraction of particulate to the matrix is maintained at similar to that of experimental result, shown in Figure 1(d). To begin with, the model was used to predict the residual plastic zone due to cooling from extrusion temperature (which is around 350 C) to room temperature. Figure 5 (a)–(c) are the contour plots showing the residual plastic strain present corresponding to the three experimental MMC samples. Figure 6(a) plots the variation of the equivalent plastic strain along the radial distance from the metal– ceramic interface into the matrix. Next on top of the residual stress the model was subjected to a tensile cyclic load with a maximum stress amplitude of 10 kPa and a time period of 1 s. From the computed results, the total energy dissipated in one cycle W as well as the maximum strain energy U was deduced. Using these values, the damping loss factor was calculated using Equation (1) and the results are shown as a bar chart in Figure 7(a). Furthermore, the analysis was repeated under the absence of residual stress condition to understand the effect of thermal cooling on the energy dissipation capability, which showed negligible energy dissipation similar to the studies of Wang et al. [12]. Next to understand the effect of particulate shape, the spherical particulate was varied to an ellipsoid, cylinder, double cone and a truncated cylinder, shown in Figure 8(a). The results from shape variation helps to understand the effect of stress concentration and aspect ratio of the particulate on the overall material behavior of the composite. Finally, the UCM containing 12 vol.% of SiC was used to study the different types of process induced defects individually, such as voids, particulate–matrix debond and particulate breakage, illustrated in Figure 8(b), and described as follows: . To understand the effect of particulate breakage, the spherical particulate was modelled as two broken pieces with no force transmitting capability in between. . To understand the effect of particulate debonding the interface between the spherical particulate and the spherical cavity of the matrix is modelled with contact elements [16] with a friction coefficient of 0.3. This enables normal hydrostatic forces to be passed to the particulate from the matrix and vice-versa while the shear stress obeys the coulombs law of friction [14]. Thus the energy dissipated due to rubbing at the interface during sliding is also taken into account. . To understand the effect of void, a spherical void of 0.0077 volume fraction was added to a 12 vol.% SiC. The void was placed at the axis of the UCM above the SiC particulate to achieve axis-symmetry in the model.

DISCUSSION The suspended beam experimental method coupled with the circle-fit approach was found to be highly repeatable and had many advantages for determining the dynamic modulus [2] and damping factor of MMC samples. Using this approach, the damping factor of pure aluminium was found to be 0.00293, which can be compared with axial damping measurements of Lazan that range from 0.0003 to 0.006 [4]. Literature review shows damping capacity of a material depends on operational frequency and strain amplitude [6,24] as well as on the difference in the material processing method which results in different microstructure in terms of grain size and shape [25]. Table 4 lists the damping factor of the monolithic sample and the three composite samples. Comparison shows that addition of SiC in the Al matrix increases the overall damping capacity and this damping improvement increases with SiC weight percentage. Based on Figure 3(a) and (b) and the experimental results listed in Table 2, the measured


1398

N. SRIKANTH ET AL.

Figure 5. Equivalent plastic strain plot in the unit cell model containing particulates with: (a) Al– 5.2 wt. % SiC; (b) Al–13.9 wt. % SiC and (c) Al–17.7 wt. % SiC.



1399

Figure 6. UCM predicted results: (a) Variation of equivalent plastic strain plot along the radial direction for different volume percentage of SiC in the Al matrix, viz., 4.4, 12.0 and 15.4%; (b) Variation of plastic zone induced damping loss factor with the number of the loading cycle in the Al–13.9 wt.% SiC; (c) Variation of equivalent plastic strain plot along the radial direction for different loading cycle in the Al–13.9 wt.% SiC.


1400

N. SRIKANTH ET AL.

Figure 7. (a) Variation of FEM predicted damping loss factor with respect to volume fraction of SiC in Al matrix; (b) comparison of FEM predicted damping loss factor for different particulate shapes and (c) Comparison of FEM predicted damping loss factor for different process induced defects.

damping results corresponds to a MMC with elongated particulates with an aspect ratio of 1.6 and possessing considerable amount of sharp edges with good interfacial bonding between the matrix and the reinforcement. It may be noted that the increase in damping of Al is realised as a result of presence and increasing wt.% of SiC irrespective of the fact that SiC has slightly lower loss factor of the order of 0.00262 [26] as compared to aluminium, which has loss factor of the order of 0.00293. Such an increase in the loss factor due to the



1401

Figure 8. (a) Particulates of similar volume but with different shape morphologies; (b) Process induced defects assumed to study their effects on the overall energy dissipation.

addition of SiC can be attributed to the SiC associated intrinsic and extrinsic damping mechanisms as described in [2]. In particulate reinforced MMCs (PRMMCs) inclusion of the hard ceramic particulates causes high residual stresses around the particulates in the form of an annular plastic zone due to large difference in the CTE of aluminium and SiC. The CTE for SiC is 4.3 ppm/ C [11] and for Al is 25.89 ppm/ C from the present study. Hence a rate dependent viscoplastic material model was used for the metal matrix behavior to accurately predict the residual plastic zone around the SiC particulate. A spheroid type shape was assumed for the SiC particulate and the residual plastic zone was predicted for composition corresponding to different volume percentage of SiC viz., 4.4, 12.0 and 15.4. Figure 5 shows the equivalent plastic strain (EQPLST) contours for the three compositions. The variation of the EQPLST is plotted in Figure 6 along the radial direction of the metallic matrix, from the metal– particulate interface. This shows clearly that maximum value of the plastic strain at the interface as well as the overall plastic zone around the particulate increases with vol.% of SiC added in the metallic matrix.


1402

N. SRIKANTH ET AL.

The FEM results can be explained by a simplistic model which is based on an elasticperfectly plastic model [15]. The size of the plastic zone Cs can be expressed as:

ET C s ¼ rs ð1 Þy

ð16Þ

where is the difference between the CTEs of Al and SiC, T is the temperature difference which is around 327 C, E and are the matrix elastic modulus and Poisson’s ratio, y is the matrix yield stress and rs is the particulate radius. Thus for the present experiment samples containing 15 mm size SiC particulates, the plastic zone radius is calculated using Equation (16) and the corresponding volume fractions are listed in Table 4. The large difference in the CTE of aluminium and SiC also leads to the production of high dislocation density at the particulate–matrix interface. Based on a prismatic dislocation-punching model of Arsenault and Shi [27], the dislocation density at a ceramic particulate is given as follows: ¼

B"Vf btð1 Vf Þ

ð17Þ

The thermal mismatch strain " is given by the difference in CTE times the temperature difference. Vf is the volume fraction of the ceramic reinforcement, b the Burgers vector, t the smallest dimension of the reinforcement (equals 15 in this experimental study) and B the geometric constant (equals 12 for equiaxed particulates). Based on Equation (17) the dislocation density for the different experiment samples were calculated, corresponding to the cooling process from 350 C to room temperature, and are listed in Table 4 using the SiC particulates’ microstructural measurements listed in Table 1 and with an assumption of 0.32 nm for the Burgers vector [28]. It is clear from Table 4 that the increase in volume fraction of reinforcement results in increase in dislocation density. Furthermore, from Table 2 it is clear that the hardness of the SiC reinforced samples is in general higher than that of unreinforced material. This difference increases with increase in wt.% of SiC. Table 2 also shows increased hardness at the interface than in the metallic matrix at a location away from the interface in any SiC containing composite, which can partly be attributed to the accumulation of residual plastic zone due to thermal mismatch at the particulate–matrix interface and finer grain size observed in the Al–SiC samples compared to that of the monolithic sample, see Table 1. The finer matrix grain size can primarily be attributed to the restricted growth of aluminium grains during solidification as a result of the presence of SiC particulates [29]. The increase in the residual plastic zone, dislocation density and reduction in grain size also leads to work hardening of the metallic matrix [30], which can also be seen in terms of increase in the yield strength change similar to that of the hardness increase [31]. Thus the changes in the mechanical properties can be seen as a confirmation to the simulation predicted features within the material. When the PRMMC is cyclically loaded, the strain " lags behind the applied stress by a phase difference . In the course of one loading period, a characteristic hysterisis loop is traced within the coordinates " (see Figure 1 (f )), whose area matches the energy dispersed in the material within one loading cycle ðW Þ. The dissipated energy contributes to heat generation while the rest is stored as energy of crystal defects accompanying plastic deformation, traditionally known as the stored energy of cold work.



1403

For metallic materials a linear thermoelasticity and linear fourier heat conduction law, the first law or energy balance equation, can be described for uniaxial stress as follows [32]: c _ k xx ¼ "_p E "_e

ð18Þ

where is the absolute temperature and , "e and "p are the components of stress, elastic strain and plastic strain, respectively, viewed as a function of coordinate x and time t. Subscripts indicate partial derivative with respect to the corresponding variable and superposed dots denote time derivative. The material constants , c, k, and E are mass density, specific heat, thermal conductivity, thermal expansion coefficient and Young’s modulus, respectively. The fraction of rate of plastic work dissipated as heat ‘’ is often assumed to be a constant parameter of 0.9 for metallic materials [32]. If an adiabatic condition prevails due to high frequency of vibration and the thermoelastic heating is negligible compared with the thermoplastic heating the above equation simplifies as follows: "_ _ ¼ c

p

ð19Þ

Studies have shown that when a material is stressed in a reversible adiabatic process, thermal gradient is induced in accordance with the Thomson effect [6]. Heat conducts from the high temperature regions to the low-temperature regions and as a consequence of the second law of thermodynamics, entropy is produced which is manifested as a conversion of useful mechanical energy into heat. This phenomenon is known as the elasto-thermodynamic (ETD) damping. In the case of flexural vibration of beams, the ETD arises due to irreversible heat transfer from the hotter region where compressive stress prevails to the cooler region where tensile stress exists. In the present study, using proper prediction models of the ETD described in [2] and the thermo-mechanical properties of the three different composite compositions, the magnitude of the ETD is found to be of the order of 104 due to the beam dimensions, which does not seem to be significant to explain the experimental observation. Considering the MMC as a multiphase system, the mechanical loss can be written as the sum of the contributions from the particulates, the general metallic matrix far from the matrix–particulate interface and the micro-plastic zones around the reinforcement, which are denoted with the subscript r, m, and zp, respectively as follows: mmc ¼ tan r þ m þ

fzp Emmc W 2 max

ð20Þ

where r and m corresponds to the damping loss factor of reinforcement and the metallic matrix, respectively, fzp is the volume fraction of the micro-plastic zone around the reinforcement, Emmc is the elastic modulus of the MMC, max is the maximum stress amplitude and W is the energy dissipated in one cycle under the presence of the microplastic zones. Thus it is clear from Equation (20) that the damping loss factor increases proportionally with the plastic zone volume fraction. The deformation of essentially all metals is, to a certain extent, time-dependent. This dependence, however, becomes more pronounced at temperatures exceeding a third of the material’s melting point. Hence it is termed viscoplastic material behavior, which refers to


1404

N. SRIKANTH ET AL.

the mechanical response of solids involving time-dependent, irreversible (inelastic) strains. In the present study, W was calculated using the FEM based UCM which mimics a rate dependent viscoplastic model (called Anand model [16] explained in a previous section) for the aluminium matrix. In the UCM the SiC particulate was assumed to possess a spheroid shape so as to achieve a conservative result in terms of energy dissipation and stiffness since it possess a minimum stress concentration and a unit aspect ratio. From the computed W and U, the plastic zone induced damping loss factor was deduced. Figure 6(b) shows the variation of the damping loss factor due to the micro-plastic zones with the number of the loading cycle which shows that the predicted damping loss factor is significant at the first cycle and reduces significantly for the second cycle and later stabilizes for the further loading cycles. Similar observation of damping variation was seen with loading cycles in the elastic-plastic based damping studies of Xu and Schmauder [11]. This can be explained by Figure 6(c), which shows the variation of the equivalent plastic strain within the unit cell increases for the first cycle and strain hardens the metallic matrix, which reduces further increase in the overall plasticity with similar loading cycles. Table 4 lists the FEM predicted damping loss factor under similar volume conditions for the three vol.% of SiC viz., 4.4, 12.0 and 15.4 which increases in the order of 27, 61 and 71%, respectively, when compared against the monolithic aluminium sample, refer Figure 7 (a). This can be explained based on Equation (20) that the damping depends directly on the volume fraction of the plastic zone under similar loading condition. Also it is clear that under practical situation, the overlapping of plastic zones can occur for situations when the plastic zone is larger with smaller inter-particulate distance, which is possible as the vol.% increases, and hence the damping increase may not be linear. In reality, SiC particulates in the MMC as seen in Figure 3(a) and (b), possess shapes similar to a polyhedron with multiple sharp edges leading to stress concentrations. To understand the effect of stress concentration on energy dissipation, the above analysis was repeated for different particulate shapes, such as ellipsoid, cylinder, truncated-cylinder and double cone. The analysis corresponds to a MMC composition of 12 vol.% SiC in Al matrix. The results were compared against a spherical particulate condition. Comparison of plastic strain contours of particulates with unit aspect ratio such as cylinder, double cone and truncated cone shown in Figure 9(a)–(c), against that of a spherical particulate assumption shown in Figure 5(b), illustrates that the maximum equivalent plastic strain depends on the presence of the stress concentration features of the particulate’s shape such as sharp edges. Comparison of spherical particulate’s plastic strain contour plot against that of ellipsoid, see Figures 5(b) and 9(d), shows that the ellipsoid has a maximum plastic strain twice that of the sphere which can be attributed to the higher aspect ratio of the ellipsoid which is twice that of the spheroid. Using the UCM predicted values of W and U, under a cyclic stress of 10 kPa amplitude with the presence of residual stress, the damping loss factor for the different particulate shapes were computed and are compared in Figure 7(b). Comparison of ellipsoid results with that of sphere shows that higher aspect ratio results in increased energy dissipation. Furthermore, the truncated cylinder provides higher energy dissipation among the similar unit aspect ratio type particulates. This may be attributed to the higher number of sharp edges present in the particulate. Further numerical work is in progress to understand and generalize such an increase in energy dissipation capability in relation to the particulate shape, size and orientation. It is clear from Equation (20), that the particulate which provides larger plastic zone (in terms of fzp) results in increased energy dissipation capability.



1405

Figure 9. Equivalent plastic strain plot in the unit cell model containing particulates with: (a) spherical shape; (b) double cone shape; (c) cylindrical shape and (d) ellipsoidal shape with an aspect ratio of 2.

To identify the presence of microstructural defects in the composites, the stiffness or the elastic modulus is a good parameter especially at low strain amplitude, similar to the present experiment. Hence the first step is to make a conservative prediction of the elastic modulus of the MMC. Among the various theoretical models [33] to predict the elastic modulus of the MMCs, the Eshelby model is the most commonly used by the researchers. It is based on the assumption of non-elastic strain (also called eigen strain) in an infinite elastic body due to the material property mismatch between the matrix and the reinforcement phase. Clyne and Withers [33] have derived an expression of the elastic modulus, Eeshelby, of the MMC based on a simplistic representation of the Eshelby model for determining a spherical particulate type system as follows: EEshelby ¼

1 ð1=Em Þ þ f ððKm Kf Þ=3Kf Þð1=8Gm Þ þ ððGm Gf Þ=Gf Þð1=3Gm Þ

ð21Þ

where Km and Kf are the bulk modulus of the matrix material and the reinforcement, respectively, while Gm and Gf are the shear modulus of the matrix and the reinforcement, respectively, when subjected to an axial loading condition. Table 4 lists the theoretical elastic modulus values that can be compared against the experimental values obtained from tensile test listed in Table 3. The discrepancy in stiffness between the experiment results compared to the theory can be mainly attributed to the process induced defects


1406

N. SRIKANTH ET AL.

such as presence of debonding at the metal–ceramic interface, presence of broken particulates in the MMC as well as due to the presence of voids [19,34]. From a damping perspective, it is interesting to note that for both good and bad interfacial bonding condition between the ceramic particulate and the metal matrix, the overall damping characteristics increases in a MMC due to different damping mechanisms. In the present study, fracture surface of the tensile test specimens for the three different MMC samples showed fracture path passing through the particulate as well as debonding at the particulate–matrix interface. The interface damping is attributed to the frictional energy loss during cyclic loading at the particulate–matrix interface. An analytical treatment on interfacial damping in particulate-reinforced MMCs was provided by Zhang et al. [34] as follows: debond ¼

3 C kf 2

ð22Þ

where is the coefficient of friction between the SiC on an Al surface, k is the stress concentration factor at an interface between a soft particulate and metal matrix, C is a correction factor and f is the volume fraction of SiC added. Thus it can be seen that the energy dissipated depends directly on the friction coefficient, stress concentration and the volume fraction of debonded particulates. In the present study, the effect of debonding on the overall damping characteristics of the MMC was computed by investigating the energy dissipated in the UCM under a debonded metal–particulate interface condition with a spherical particulate morphology. Analysis was performed for a friction coefficient of 0.3, which showed that the damping loss factor of the Al containing 12 vol.% SiC is of the order of 3.811E-03. This when compared against a fully bonded interfacial condition, which is around 2.592E-03, shows clearly that the damping loss factor has increased by 47%, as shown in Figure 7(c). Figure 10(a) shows the equivalent plastic strain contours in the unit cell under the debonded metal–particulate interface condition. Next an analysis was performed for a broken particulate condition, which showed a reduction in stiffness arises since the forces do not pass through the particulate thereby the strain energy decreases compared to the fully bonded particulate. Figure 10(b) shows the equivalent plastic strain contours in the unit cell under the broken particulate condition. It is clear that the plastic strain is more localized in the metallic matrix at the crack tip. This leads to an increase in plastic energy dissipation which along with the stiffness drop results in an increase of the damping loss factor by a factor of 31% compared to the ideal condition. Further, presence of void was studied which illustrated that the void alters the plastic strain contours and similarly results in both reduction in strain energy and an increase in plastic work done. Figure 10(c) shows the equivalent plastic strain contours in the unit cell when a void fraction of 0.0077 is present in the Al-12 vol.% SiC. According to Fougere et al. [35] the elastic moduli ‘Ep’ of porous materials depend on the porosity percentage ‘p’ by an exponential function, Ep ¼ E0 expðpÞ, where is a material constant which ranges between 2 and 4 and E0 is the Young’s modulus under no porosity condition. Thus a proportional drop in strain energy is expected. Comparison of results in Figure 7(c), corresponding to with and without void presence in the UCM, shows a minimal increase in the damping loss factor of 1%. This matches well with research findings of Zhang et al. [36] which showed that the presence of voids of few volume percentage in the Al matrix provides marginal increase in the overall damping capacity of the metallic matrix.



1407

Figure 10. Equivalent plastic strain plot in the unit cell model under the presence of: (a) debonded metal–particulate interface; (b) a broken particulate along the central half plane (at y ¼ 0) and (c) void of 0.77 vol.%.


1408

N. SRIKANTH ET AL.

Figure 11. Variation of plastic zone induced damping coefficient with strain amplitude.

The present study can be easily extended to higher strain amplitude when tested under the conditions explained in [11], to explain the experimental observation of Zhang et al. [36]. Figure 11 shows that the present model predicts an increase in micro-plasticity induced energy dissipation with the increase in applied strain amplitude. This matches with the elastic-plastic model prediction of Xu and Schmauder [11]. Based on the results of the microstructural studies, refer Table 1, conducted on the actual experiment samples, the reinforcement are elongated particulates with enough sharp edges. On the contrary, the present FEM model assumes a spherical reinforcement and a uniform inter-particulate distance, since it is based on a unit cell concept, which may not be true due to the presence of clusters observed in the SEM micrograph, shown in Figure 3(a). Based on the study of Zhou et al. [37], presence of clusters leads to unequal load sharing and overlapping of the thermally induced plastic zones. Secondly, the model assumes an ideal interface between the SiC and the Al matrix, which is not realistic, based on the microstructural studies. However, the model can account to some extent sharp faceted particulate shapes compared to the spherical shape assumed in the present study. Further in terms of interfacial condition, both fully-bonded and a fully-debonded interface between the SiC and the Al-matrix were assumed in the FEM model which may be considered as the two extremes of the real situation. In reality, the experimental sample would be expected to have both bonded and debonded interfaces as well as partial-bonded interfaces. While further efforts are being made to circumvent these shortcomings, the results of the modelling follow very closely to the experimental results and thus promises to be a useful design tool to provide conservative prediction during the initial phase of the material selection for the composite design.

CONCLUSIONS The following inferences can be made from the analysis: 1. The FEM based cell model used in the present study enables the determination of damping loss factor of composites due to the presence of residual plastic zones induced



2. 3.

4.

5.

1409

due to thermal mismatch during cooling from extrusion temperature to room temperature. The results matches well with experimental results obtained from impact based suspended beam experiments, which corresponds to low strain amplitude condition. Numerical results also show similar increasing trend in damping loss factor with weight percentage of SiC added in the metallic matrix consistent with the experimental observation of suspended beam experiments. Effect of stress concentration due to sharp faceted particulates increases the energy dissipation capability of the metallic matrix due to the increased presence of plastic zone. The present model is capable to study the effect of process induced defects such as particulate breakage, presence of void in the metallic matrix and debonding at the metal–particulate interface on the overall energy dissipation capability of the MMC. Comparison against ideally bonded condition shows that these defects increase the energy dissipation capability of the MMC.

REFERENCES 1. Lavernia, E.J., Perez, R.J. and Zhang, J. (1995). Metallurgical and Materials Transactions A, 26A: 2803–2818. 2. Srikanth, N., Lim, S.C.V. and Gupta, M. (2002). Journal of Composite Materials, 36(20): 2339–2355. 3. Srikanth, N. and Gupta, M. (2001). Scripta Materialia, 45: 1031–1037. 4. Lazan, B.J. (1968). Damping of Materials and Members in Structural Mechanics, Pergamon Press, New York. 5. Goodman, L.E. (1994). Material Damping and Slip Damping, Shock and Vibration Handbook, In: Harris, C.M. (ed.), 3rd edn, Chapter 36, pp. 1–27. 6. Kinra, V.K. and Wolfenden, A. (1993). M3D: Mechanics and Mechanisms of Material Damping, ASTM STP. 1169, American Society of Testing Materials, Philadelphia. 7. Llorca, J. (1995). Acta Metall. Mater., 42: 151–162. 8. Bao, G. (1992). Acta Metall. Mater., 40: 2547–2555. 9. Shen, Y.L., Finot, M., Needleman, A. and Suresh, S. (1995). Acta Metall. Mater., 43: 1701–1722. 10. Farrissey, L., Schamauder, S., Dong, M., Soppa, E., Poech, M.H. and McHugh, P. (1999). Computational Materials Science, 15: 1–10. 11. Xu, D. and Schmauder, S. (1999). Computational Materials Science, 15: 96–100. 12. Wang, J., Zhang, Z. and Yang, G. (2000). Computational Materials Science, 18: 205–211. 13. Shee, S.K., Pradhan, S.K. and De, M. (1998). Materials Chemistry and Physics, 52: 228–235. 14. Cook, R.D., Malkus, D.S. and Plesha, M.E. (1989). Concepts and Applications of Finite Element Analysis, 3rd edn, John Wiley and Sons, New York. 15. ANSYS Reference-Theory Manual, (1998). Version 5.5, Swanson Analysis Systems Publications, Houston, U.S.A. 16. Anand, L. (1995). Int. J. Plasticity, 1: 213–251. 17. Brown, S.B., Kim, K.H. and Anand, L. (1989). Int. J. Plasticity, 5: 95–130. 18. Srikanth, N. and Gupta, M. (2002). Damping Characterization of Aluminium Based Composites Using an Innovative Circle-fit Approach, Key Engineering Materials, 227: 211–216. 19. Srikanth, N., Tham, L.M. and Gupta, M. (1999). Alum. Trans., 1: 11–18.


1410

N. SRIKANTH ET AL.

20. ASTM E8M-96 (1996). Standard Test Methods of Tension Testing of Metallic Materials, Annual Book of ASTM Standards, American Society for Testing and Materials, Philadelphia. 21. ASTM C1259-98 (1998). Dynamic Young’s Modulus, Shear modulus, and Poisson’s Ratio for Advanced Ceramics by Impulse Excitation of Vibration, Annual Book of ASTM Standards, American Society for Testing and Materials, Philadelphia. 22. Ewins, D.J. (1984). Modal Testing: Theory and Practice, Research Studies Press, New York. 23. Teixeira-Dias, F. and Menezes, L.F. (2001). Computational Materials Science, 21: 26–36. 24. Wolfenden, A. and Wolla, J.M. (1989). Journal of Materials Science, 24: 3205–3212. 25. Keˆ, T.S. (1947). Physical Review, 71: 533–546. 26. Thirumalai, R., and Gibson, R. (1993). Damping of Multiphase Inorganic Materials, In: Bhagat, R.B. (ed.), Materials Park, ASM International, Ohio. 27. Minoro, T. and Arsenault, R.J. (1989). Metal Matrix Composites – Thermomechanical Behavior, Pergamon Publishers, New York. 28. Frost, H.J. and Ashby, M.F. (1982). Deformation-Mechanism Maps: the Plasticity and Creep of Metals and Ceramics, Pergamon Press, Oxford, New York. 29. Lou, A. (1995). Metallurgical and Materials Transactions A, 26A: 2445–2451. 30. Niklas, A., Froyen, L., Delaey, L. and Buekenhout, L. (1991). Materials Science and Engineering, A135: 225. 31. Meyers, M.A. and Chawla, K.K. (1999). Mechanical Behavior of Materials, Prentice Hall International Inc., USA. 32. Hodowany, J., Ravichandran, G., Rosakis, A.J. and Rosakis, P. (2000). Experimental Mechanics, 40(2): 113–123. 33. Clyne, T.W. and Withers, P.J. (1993). An Introduction To Metal Matrix Composites, Cambridge University Press, Australia. 34. Zhang, J., Perez, R.J. and Lavernia, E.J. (1994). Acta Metall. Mater., 42(2): 395–409. 35. Fougere, G.E., Riester, L., Ferber, M., Weertman, J.R. and Siegel, R.W. (1995). Mater. Sci. Eng. A, 204: 1–6. 36. Zhang, J., Perez, R.J. and Lavernia, E.J. (1993). Journal of Materials Science, 28: 835–846. 37. Zhou, Y.C., Long, S.G., Duan, Z.P. and Hashida, T. (2001). Journal of Engineering Materials and Technology, 123: 251–260.
